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% 3: Design Methodology
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The design and optimization of \DSms through hybridized evolutionary strategies is the
primary goal of this work. Specifically, a hybrid orthogonal genetic algorithm is proposed
to optimize the double input, single output IIR filter design which characterizes \DSm
functionality. This algorithm is shown to be robust with only minimal steady-state
misadjustment. Further, this design methodology is shown to outperform established
classical and contemporary design methods.

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%%% Design Methodology
\section{Hybrid Orthogonal Evolutionary Strategy (HOES)}
The Hybrid Orthogonal Evolutionary Strategy (HOES) is quite similar to traditional
evolutionary strategies in that the operating principle of evolving the 'most fit'
individual over successive generations remains unchanged. However, in contrast to
traditional evolutionary strategies, the evolutionary process will not be restricted to
pure stochastic transfer of genetic information between progenitors. The process will be
'hybridized' by intelligently sharing genetic information during reproduction such that
the progeny is optimal given the available genotype. Note that this optimality is defined
by context of the fitness function used to evaluate the population. This hybridization of
the evolutionary process is precisely what allows for faster, more accurate, and more
robust filter designs to be evolved.

Figure \ref{fig:HOES_flow_chart} illustrates the flow of the HOES. Each stage of the HOES
will be addressed in the following sections.

\begin{figure}[htbp]
\centering
\includegraphics[height=7.5in]{./working_figures/HOES_flow.eps}
\caption{Hybrid Orthogonal Evolutionary Strategy (HOES) Flow Chart}
\label{fig:HOES_flow_chart}
\end{figure}

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%%% Chromosome Representation
\input{chromosome_representation}

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%%% Population Initialization
\input{population_initialization}

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%%% Fitness Evaluation
\subsection{Fitness Evaluation}
Each member of a population is evaluated for relative fitness during each generation 
(iteration). The result of this fitness evaluation is what drives the evolution of the
greater population. Based on the relative fitness, 'more fit' individuals have an
increased likelyhood of mating and producing offspring thereby preserving their genetic
information in the population genotype. Conversely, 'less fit' individuals have a
decreased likelyhood of mating thereby insuring the elimination of their genetic
information from the population genotype. This process of natural selection according to
relative fitness is central to evolutionary strategies \cite{darwin_origin_2001}.

The algorithm itself is application agnostic and independent of the problem space.
As such, the fitness function (objective function) fully characterizes the problem
space. The objective function discussed here is analogous to the cost function for the
optimization algorithms previously presented. It is the authors opinion that the
evolutionary strategy and the objective function are mutually exclusive. The challenge of
objectifying a particular problem space is wholly independent of the challenge of
architecting an evolutionary strategy.

The various objective functions implemented for the purpose of this work will be
presented in the following sections. However, they are all alike in that they seek to
minimize the result of objective function evaluation (or cost). As such, the HOES evolves
a population whose most fit chromosome represents the global minimum for the respective
objective function. Further, the problem space is generally constrained through the
penalization of chromosomes which evolve outside of the established constraints. The
constraints and respective penalties will also be addressed in the following sections.

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%%% Linear-Ranking Selection
\input{linear_ranking_selection}

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%%% Single-Point Crossover Operation
\input{single_point_crossover}

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%%% Taguchi Method
\input{taguchi_method}

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%%% Point-Swap Mutation
\subsection{Point-Swap Mutation}
'doc stub'

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%%% Convergence
\subsection{Convergence}
'doc stub'

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%%% DTDSM Design
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\section{Discrete-Time \DSM Design}

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%%% DTDSM Design: LPNTF
\subsection{Low-Pass NTF Objective Function}'doc stub'
\subsubsection{Initialization}'doc stub'
\subsubsection{Constraints and Penalties}'doc stub'
\subsubsection{Cost Function Characterization}'doc stub'

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%%% DTDSM Design: LPSTF
\subsection{Low-Pass STF Objective Function}'doc stub'
\subsubsection{Initialization}'doc stub'
\subsubsection{Constraints and Penalties}'doc stub'
\subsubsection{Cost Function Characterization}'doc stub'

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%%% DTDSM Design: VLIF (Bandpass)
\subsection{VLIF Band-Pass Designs}
'doc stub'

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%%% DTDSM Design: Output Decimation
\subsection{Output Decimation}
'doc stub'

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%%% DTDSM Design: Simulation
\subsection{Simulation}
'doc stub'

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%%% CTDSM Design
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\section{Continuous-Time \DSM Design}
'doc stub'

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%%% CTDSM Design: LPNTF
\subsection{Low-Pass NTF Objective Function}'doc stub'
\subsubsection{Initialization}'doc stub'
\subsubsection{Constraints and Penalties}'doc stub'
\subsubsection{Cost Function Characterization}'doc stub'

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%%% CTDSM Design: LPSTF
\subsection{Low-Pass STF Objective Function}'doc stub'
\subsubsection{Initialization}'doc stub'
\subsubsection{Constraints and Penalties}'doc stub'
\subsubsection{Cost Function Characterization}'doc stub'

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%%% CTDSM Design: VLIF (Bandpass)
\subsection{VLIF Band-Pass Designs}
'doc stub'

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%%% CTDSM Design: Output Decimation
\subsection{Output Decimation}
'doc stub'

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%%% CTDSM Design: Simulation
\subsection{Simulation}
'doc stub'